International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 37-48 | Cite as

On the Galois Lattice of Bipartite Distance Hereditary Graphs

  • Nicola Apollonio
  • Massimiliano Caramia
  • Paolo Giulio FranciosaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)


We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. We show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal time computation of neighborhood intersections and maximal bicliques.


Galois lattice Transitive reduction Distance hereditary graph Ptolemaic graph 


  1. 1.
    Amilhastre, J., Vilarem, M.C., Janssen, P.: Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs. Discrete Appl. Math. 86, 125–144 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Atkinson, M.D.: On computing the number of linear extensions of a tree. Order 7, 23–25 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Combin. Theory Ser. B 41, 182–208 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Belohlavek, R., De Baets, B., Outrata, J., Vychodil, V.: Trees in concept lattices. In: Torra, V., Narukawa, Y., Yoshida, Y. (eds.) MDAI 2007. LNCS (LNAI), vol. 4617, pp. 174–184. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  5. 5.
    Berry, A., Sigayret, A.: Dismantlable lattices in the mirror. In: Cellier, P., Distel, F., Ganter, B. (eds.) ICFCA 2013. LNCS, vol. 7880, pp. 44–59. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  6. 6.
    Brucker, F., Gély, A.: Crown-free lattices and their related graphs. Order 28(3), 443–454 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cornelsen, S., Di Stefano, G.: Treelike comparability graphs. Discrete Appl. Math. 157, 1711–1722 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fagin, R.: Degrees of acyclicity for hypergraphs and relational database schemes. J. ACM 30(3), 514–550 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ganter, B., Wille, R.: Formal Concept Analysis - Mathematical Foundations. Springer, Heidelberg (1999)zbMATHCrossRefGoogle Scholar
  10. 10.
    Howorka, E.: A characterization of distance-hereditary graphs. Q. J. Math. 2(26), 417–420 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Howorka, E.: A characterization of Ptolemaic graphs, survey of results. In: Proceedings of the 8th SE Conference Combinatorics, Graph Theory and Computing, pp. 355–361 (1977)Google Scholar
  12. 12.
    Peled, U.N., Wu, J.: Restricted unimodular chordal graphs. J. Graph Theory 30(2), 121–136 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Rival, I.: Lattices with doubly irreducible elements. Can. Math. Bull. 17(1), 91–95 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Schieber, G., Vishkin, U.: On finding lowest common ancestors: simplification and parallelization. SIAM J. Comput. 17(6), 1253–1262 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Swaminathan, R.P., Wagner, D.B.: The arborescence-realization problem. Discrete Appl. Math. 59, 267–283 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Syslo, M.M.: Series-parallel graphs and depth-first search trees. IEEE Trans. Circuits Syst. 31(12), 1029–1033 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore, Maryland (1992)zbMATHGoogle Scholar
  18. 18.
    Whitney, H.: 2-isomorphic graphs. Am. Math. J. 55, 245–254 (1933)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Nicola Apollonio
    • 1
  • Massimiliano Caramia
    • 2
  • Paolo Giulio Franciosa
    • 3
    Email author
  1. 1.Istituto per le Applicazioni del Calcolo M. Picone, CNRRomeItaly
  2. 2.Dipartimento di Ingegneria dell’ImpresaUniversità di Roma “Tor Vergata”RomeItaly
  3. 3.Dipartimento di Scienze StatisticheSapienza Università di RomaRomeItaly

Personalised recommendations